Upper bounds on Nusselt number at finite Prandtl number

We study Rayleigh B'enard convection based on the Boussinesq approximation. We are interested in upper bounds on the Nusselt number $\mathrm{N}\mathrm{u}$, the upwards heat transport, in terms of the Rayleigh number $\mathrm{R}\mathrm{a}$, that characterizes the relative strength of the driving mechanism and the Prandtl number $\mathrm{P}\mathrm{r}$, that characterizes the strength of the inertial effects. We show that, up to logarithmic corrections, the upper bound $\mathrm{N}\mathrm{u}\lesssim {\mathrm{R}\mathrm{a}}^{\frac{1}{3}}$ of Constantin and Doering in 1999 persists as long as $\mathrm{P}\mathrm{r}\gtrsim {\mathrm{R}\mathrm{a}}^{\frac{1}{3}}$ and then crosses over to $\mathrm{N}\mathrm{u}\lesssim {\mathrm{P}\mathrm{r}}^{-\frac{1}{2}}{\mathrm{R}\mathrm{a}}^{\frac{1}{2}}$. This result improves the one of Wang by going beyond the perturbative regime $\mathrm{P}\mathrm{r}\gg \mathrm{R}\mathrm{a}$. The proof uses a new way to estimate the transport nonlinearity in the Navier-Stokes equations capitalizing on the no-slip boundary condition. It relies on a new Calder'on-Zygmund estimate for the non-stationary Stokes equations in $L^1$ with a borderline Muckenhoupt weight.